Question

Express `x - 1/x = 3` as a quadratic equation in standard form and hence find its roots. Also, find the value of ‘a’ for which the equation `x + 1/x = a`, when expressed as a quadratic equation, has real and equal roots.

Answer 1

`x - 1/x = 3`

⇒ `(x^2 - 1)/x = 3`

⇒ x² – 1 = 3x

⇒ x² – 3x – 1 = 0

⇒ x² + (–3)x + (–1) = 0

a = 1, b = –3, c = –1

`x = (-b ± sqrt(b^2 - 4ac))/(2a)`

= `(3 ± sqrt(9 + 4))/2`

= `(3 ± sqrt(13))/2`

`x = (3 + sqrt(13))/2, (3 - sqrt(13))/2`

a = ?

`x + 1/x = a`

⇒ `(x^2 + 1)/x = a`

⇒ x² + 1 = ax

⇒ x² – ax + 1 = 0

⇒ x² + (–a)x + 1 = 0

For real and equal roots

b² – 4ac = 0

a = 1, b = –a, c = 1

b² – 4ac = 0

⇒ (–a)² – 4 × 1 × 1 = 0

⇒ a² – 4 = 0

⇒ a² = 4

⇒ `a = sqrt(4)`

⇒ a = ± 2